Exclusive and Independent Probability Events

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There are two commonly used terms in Probability and generically in Statistics:

Exclusive or disjoint events.

Independent events.

A great number of theorems and applications within the Statistics field depend on whether the studied events are either mutually exclusive or not, and if they are either mutually independent or not as well.

Disjoint or mutually exclusive events

Two events are disjoint if they cannot occur at the same time. For instance, the age ranges probabilities for a customers are disjoint. It cannot occur simultaneously that a particular customer is more than twenty and less than twenty year old.

Other example is the status of an order. It may be in preparation, at the magazine, en route or delivered to the consignee; being those states mutually exclusive as well.

On the other hand non-disjoint events may coexist at the same point in time. A customer may live in a particular town and be at concurrently more than twenty year old. Those two conditions are not mutually exclusive. Those type of events are not disjoint or mutually exclusive. In the same way, an order may either be in preparation and being assembled or in preparation and ready for delivery all together.

Depending if two or more events are or not disjoint, the way to calculate their probabilities is different. And the outcome of the probabilistic calculus will vary therefore based on it.

Dependent events

Two events are independent when the outcome of one does not depend on the other. In terms of probability, two events are independent when the probability of one of them is not affected by the probability of the other event.

This is the case of the games of chance like lotteries and casinos. Every time the die is rolled the chances to obtain a particular outcome do not change; at each roll the probability of obtaining any of the six possible values for a six-sided die is equal to \(\)\(\frac{1}{6}\).

Conversely, dependent events are affected by their respective probabilities. In this case we talk about conditional probability and that probability is expressed using the nomenclature \(P(A|B)\). An example may be the probability of selling an on-line product when the user has already opened an account on the site and returns. It is different if the second event (account opened previously) occurs or not.